Question

The equation of the curve whose slope at any point is equal to $y+2x$ and which passes through the origin is

Answer 1

Given slope of curve at any point $=y+2x$

i.e. $\frac{dy}{dx}=y+2x$

$\Rightarrow \frac{dy}{dx}-y=2x$ ....(1)

It is of the form $\frac{dy}{dx}+Py=Q$

Here, $P=-1,Q=2x$
$I.F.=e^{\int pdx}=e^{\int -1dx}=e^{-x}$

$\therefore$ The solution of eqn (1) is given by

$y{e}^{-x}=\int 2x{e}^{-x}dx+c$
$=2[x\left({-e}^{-x}\right)-\int 1.(-e^{-x})dx]+c$
$=2[-x{e}^{-x}-e^{-x}]+c$
$\Rightarrow y=-2x-2+{ce}^x$
which is the equation of the family of curves.
Since, it passes through the origin $(0,0),$
$0=-2+c$ or $c=2$
$\therefore$ The required equation of the curve is $2x+y+2={2e}^x$